To solve an optimization problem, express the quantity to be maximized or minimized as a function of one variable, find its critical points, and compare all valid candidates, including domain endpoints. The final answer must be interpreted in the problem's context.
The Optimization Method
This is SL 5.7: Optimisation in IB Mathematics: Applications and Interpretation.
- Identify the objective function, such as profit or cost .
- Use any given constraint to write it as a function of one variable.
- State the realistic domain.
- Find stationary points by solving or , analytically or using a GDC.
- Test each stationary point and the domain endpoints.
- Interpret the optimum with appropriate units and accuracy.
| Required result | Evidence that confirms it |
|---|---|
| Maximum profit | Derivative changes from positive to negative, or the value exceeds all endpoint values |
| Minimum cost | Derivative changes from negative to positive, or the value is below all endpoint values |
For profit problems, profit equals revenue minus cost. If represents whole items, a continuous optimum may not be an integer, so compare neighboring feasible integers; here is already valid. Cost minimization uses the same method, but selects the lowest candidate value. With a GDC, record the equation or graph used before the numerical result.
Worked Example: Maximizing Profit
A company sells items, where . Its revenue and cost are
Therefore, the profit function is
Differentiate and solve:
Since , this stationary point is a maximum. Checking the candidates gives
Hence, the maximum profit is monetary units, achieved by selling items.
Exam Technique
A common misconception is that every stationary point is automatically the required optimum. Examiners expect the objective function, the equation , a valid maximum or minimum test, endpoint checks where relevant, and a contextual conclusion. On every Math AI paper, a GDC is available, but show the mathematical set-up rather than only a calculator answer.
